Calculus Examples

$\int \sin (x) (4 \csc (x) - \cot (x)) d x$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int \sin (x) (4 \csc (x)) + \sin (x) (- \cot (x)) d x$

Step 1.2

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.2.1

Reorder $\sin (x)$ and $4 \csc (x)$ .

$\int 4 \csc (x) \sin (x) + \sin (x) (- \cot (x)) d x$

Step 1.2.2

Add parentheses.

$\int 4 (\csc (x) \sin (x)) + \sin (x) (- \cot (x)) d x$

Step 1.2.3

Rewrite $\sin (x) (4 \csc (x))$ in terms of sines and cosines.

$\int 4 (\frac{1}{\sin (x)} \sin (x)) + \sin (x) (- \cot (x)) d x$

Step 1.2.4

Cancel the common factors.

$\int 4 \cdot 1 + \sin (x) (- \cot (x)) d x$

Step 1.3

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.3.1

Reorder $\sin (x)$ and $- \cot (x)$ .

$\int 4 \cdot 1 - \cot (x) \sin (x) d x$

Step 1.3.2

Add parentheses.

$\int 4 \cdot 1 - (\cot (x) \sin (x)) d x$

Step 1.3.3

Rewrite $\sin (x) (- \cot (x))$ in terms of sines and cosines.

$\int 4 \cdot 1 - (\frac{\cos (x)}{\sin (x)} \sin (x)) d x$

Step 1.3.4

Cancel the common factors.

$\int 4 \cdot 1 - \cos (x) d x$

Step 1.4

Multiply $4$ by $1$ .

$\int 4 - \cos (x) d x$

Step 2

Split the single integral into multiple integrals.

$\int 4 d x + \int - \cos (x) d x$

Step 3

Apply the constant rule.

$4 x + C + \int - \cos (x) d x$

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$4 x + C - \int \cos (x) d x$

Step 5

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

$4 x + C - (\sin (x) + C)$

Step 6

Simplify.

$4 x - \sin (x) + C$